Abstract : The representation formats and behaviors of floating point arithmetics available in computers are defined by the IEEE-754 standard. This standard imposes the system to return as a result of one of the four basic operations (+, *, /, sqrt), the rounding of the exact result. This property is called <>,this warranties the quality of the result. It enables construction of proof that this particular algorithms can be manipulated independently of the machine. However, due to the <>, elementary functions (sine, cosine, exponential...) areabsent in the IEEE-754 standard. Contrary to basic operations, it is difficult to discover the necessary accuracy required to guarantee correct rounding for elementary functions. However if therepresentation format is set, it is possible that an exhaustivesearch will help determine this bound: it was Lefevre's work for the double precision.

The objectives of this thesis is to exploit these bounds for eachfunctions and rounding modes, to certify correct rounding in doubleprecision. Thanks to this bound we have defined an evaluation within 2 steps: a quick phase which is based on the property of the IEEE standard that often proves satisfactory and an accurate step based on multiprecision operations which is precise all the time. For the second step we have designed a multiprecision library which was optimized in order to acquire precision corresponding to the bound, and the caracteristics of processors in 2003.